This OPE task, as pointed in out in Figure REF  (yellow box on top right), is a retrospective OPE task at the population-level.
The most common example of this is the expected causal Effect of the Treatment on the Treated or ETT [1]}, denoted by \(\mathbb {E}[Y_t^{A_t=a_t^{\prime }}|A_t=a_t]\)  (where \(a_t\)  is the original medication and \(a_t^{\prime }\)  is the changed medication. Specifically, this evaluates the expected outcome had the patients been treated with \(a_t^{\prime }\)  instead of \(a_t\) . Note that this is an example of a hard intervention. The estimand is explicitly conditioning on evidence that \(A_t=a_t\)  from observational data, but counterfactually allowing the treatment to vary (to \(a_t^{\prime }\) ) to reason about the change in the mean. Another example, in the case of soft interventions, is when one may be interested in understanding the expected outcome using policy \(\pi _e\)  having observed outcomes using policy \(\pi _b\)  retrospectively, i.e. for the same set of patients treated with \(\pi _b\)  denoted as: \(\mathbb {E}[Y_T^{\pi _e}({\mathbf {U}})|Y_T^{\pi _{b}}]\) . We show the non-identifiability of ETT as an example here in the binary setting (note that this is a parametric assumption) as derived in [2]} for the case where \(Y_t\)  and \(A_t\)  are binary in Equation REF . Note that Equation REF  can be written using law of total expectation. Then the target causal query is given by Equation REF .
\(\begin{split}\mathbb {E}[Y_t^{A_t=a_t^{\prime }}|A_t=a_t] = \frac{\mathbb {E}[Y_t^{A_t=a_t^{\prime }}] - \mathbb {E}[Y_t^{A_t=a_t^{\prime }}|A_t=a_t^{\prime }]p(a_t^{\prime })}{p(a_t)}\end{split}\) 
